The String Coffee Table

We had been told before that the claim is that for the curved $\beta-\gamma$ system appearing in the pure spinor superstring formulation, the stress-energy tensor is locally BRST exact on patches $U_\alpha$

but that the composite operator $G$ (playing the role of the $b$-ghost in the NSR formulation of the superstring) is not globally defined, but Čech-closed only up to a BRST-exact term

A similar statement was claimed to be true for $G^{(1)}$ and so on, up to a $G^{(3)}$ which was finally globally defined.

Now Nikita Nekrasov presents some details of this computation. If you scroll precisely 1 hour, 1 minute and 28 seconds into the Quicktime video of the third of his talks, you’ll see Nikita giving the detailed formulas for the various $G$-operators as composites of the elementary worldsheet fields.

Previously I had noted that hence the formal combination

defines a cocycle for the combined operator

or rather that $T$ is a $D$-coboundary

Nikita Nekrasov does use the same formulation now in his talk (see about 1 hour, 7 minutes and 20 seconds into the video) and makes some remarks about how this suggests that one should interpret $D = \delta \pm Q$ as the ‘true’ physical BRST operator. This way, we would have that the stress-energy tensor is BRST-exact after all.

There are some vague but very nice comments in this third talk hinting at the deeper mathematical structure hiding behind this. In particular, it is emphasized how Čech cohomology and Dolbeault cohomology are related and how this can be used to reformulate physics one way or the other.

One open puzzle which gets some attention is the question why, in the above formulas, we have (depending on how you count) a 3-cocycle, i.e. why there are a total of 3 ‘Čech-potentials’ $G^{(i)}$ for $G$.

Not that I know the answer, but I note the following:

As I have mentioned before , nonabelian bundles, gerbes, etc. with connection are similarly described by cocycles in the Čech hypercohomology for certain complexes with differential $Q$. (This is the content of the last section of my thesis.) Compared to these formulations, one finds the following:

The stress-energy tensor $T$ plays the role of the curvature of the given object (bundle, 2-bundle, …)

The ‘ghost’ $G$ plays the role of the connection on the given object (an ordinary connection on a bundle, a 2-connection on a 2-bundle, etc.).

The operator $G^{(1)}$ plays the role of the transition data on double overlaps.

For an ordinary bundle that’s it. We note that bundles correspond to 1-cocycles (using the above counting).

The operartor $G^{(2)}$ plays the role of the transition data on triple overlaps.

For a gerbe or 2-bundle that’s it. Hence gerbes correspond to 2-cocycles.

The operator $G^{(3)}$ plays the role of the transition data on quadruple overlaps.

This would be the last piece of data for a 3-bundle or a 2-gerbe with connection.

Since a 3-bundle or 2-gerbe with connection is something that assigns holonomy to worldvolumes of membranes instead of to worldsheets of strings, it is maybe kind of surprising that in the pure spinor superstring one seems to find an object described by such a 3-cocycle, not a 2-cocycle.

On the other hand, maybe it is not that surprising after all, if you take into acount this… and that…